Abstract
In this paper, we study the problem of fair allocation in production economies where agents have different preferences and unequal production skills. We characterize the equal income Walrasian solution and the proportional solution using axioms of equity and a certain form of implementability developed by Yamada and Yoshihara (Int J Game Theory 36:85–106, 2007).
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Notes
SPI was firstly introduced by Yoshihara (1998).
In other environments, there are similar characterizations. Nagahisa and Suh (1995) show that, in pure exchange economies with differentiable utility functions, the only solution satisfying No Envy, Pareto Efficiency, and Local Independence is the \(\textit{EI}\) solution. Local Independence states that if an allocation is socially selected and utility functions change to retain the marginal rates of substitutions at the allocation, then the allocation should remain selected. This axiom is stronger than Maskin Monotonicity over the domain of differentiable utilities. In a model of indivisible objectives with money, Sakai (2007) characterizes the \(\textit{EI}\) solution by Equal Treatment of Equals, Maskin Monotonicity and a continuity property.
Yamada and Yoshihara (2007) justified these conditions also in terms of responsibility and compensation (Fleurbaey and Maniquet 2011). If agents are responsible for their preferences, SPI is also interpreted as a responsibility principle since it requires the allocation rule to be independent from particular changes in agents’ preferences (with technologies fixed). Note that under PE, SPI is stronger than Maskin Monotonicity, which is justified as a responsibility principle by Fleurbaey and Maniquet (1996). On the other hand, IUS requires that even if agents are not responsible for their skills, the change of nonworking agent’s skill (with the same supporting price) should not be compensated for.
In this case, ND is excluded since it implies \({ NIS}^*\).
Note that \(\textit{EI}\) is a form of \(\tilde{u}\) budget equivalent solution with a particular reference preference.
References
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The author is especially grateful to Naoki Yoshihara for helpful comments and suggestions. The author also thanks Benoit Decerf, Marc Fleurbaey, Haruo Imai, Chiaki Hara, Kazumi Hori, Ryoichi Nagahisa, Takashi Kunimoto, Z. Emel Ozturk, Marcus Pivato, Clements Puppe, Koji Takamiya, Takuma Wakayama, and audiences at Hitotsubashi University, Kyoto University, Niigata University, and New Directions in Welfare III at OECD Paris. Financial support from the Japan Society for the Promotion of Science is gratefully acknowledged.
Kaname Miyagishima—JSPS Research Fellow.
Appendix
Appendix
In this appendix, we show the independence of the axioms in the theorems. First we provide solutions satisfying all the axioms in Theorem 1 except:

(1)
PE. Let \(e' \in \mathcal {E}\) be such that for all \(\varvec{z}\in \textit{EI}(e')\), \(c_{i^*}>0\) for some \(i^* \in N\) and \(R_{i^*} \not = R_{j}\) for all \(j \not = i^*\). Define \(S^1\) as follows. For all \(e \in \mathcal {E} \backslash \{e'\}\), \(\varvec{z}\in S^1(e)\) if \(\varvec{z}\in \textit{EI}(e)\); for \(e'\), \(\varvec{z}\in S^1(e')\) if \(z_{i^*}=(l_{i^*}', c_{i^*}'  \epsilon )\) for a sufficiently small \(\epsilon >0\), \(\varvec{z}_{i} = \varvec{z}_{i}'\), where \(\varvec{z}' \in \textit{EI}(e')\).

(2)
SPI. Consider the Egalitarian Equivalent solution EE considered by Fleurbaey and Maniquet (1999, Definition 15).

(3)
IUS. Consider the \(\tilde{u}\) budget equivalent solution \(BE^{\tilde{u}}\) (other than \(\textit{EI}\)) introduced by Fleurbaey and Maniquet (1996).^{Footnote 8}

(4)
EREL. Define \(S^2\) as follows. Let \(\overline{B}(s_{i},w, b_{i})=\{z' \in [0,\overline{l}] \times \mathbb {R}_{+} c' \le ws_{i}l' + b_{i}\}\). For all \(e \in \mathcal {E}\), \(\varvec{z}\in S^2(e)\) if \(\varvec{z}\in P(e)\) and there exists \(w \in W(\varvec{z}, e)\) such that for all \(i \in N\),
$$\begin{aligned} z_{i} \in \mathop {\text{ argmax }}_{\tilde{z} \in \overline{B}(s_{i},w, b_{i})}u_{i}(\tilde{z}), \end{aligned}$$where \(b_{i} \not = b_{j}\) for all \(i,j \in N\).

(5)
ND. Consider the proportional solution.
Next we present examples of solutions satisfying all the axioms in Theorem 2 but:

(1)
PE. Let \(e' \in \mathcal {E}\) be such that for all \(\varvec{z}\in \textit{PR}(e')\), \(c_{i^*}>0\) for some \(i^* \in N\) and \(R_{i^*} \not = R_{j}\) for all \(j \not = i^*\). Define \(S^3\) as follows. For all \(e \in \mathcal {E} \backslash \{e'\}\), \(\varvec{z}\in S^3(e)\) if \(\varvec{z}\in \textit{PR}(e)\); for \(e'\), \(\varvec{z}\in S^3(e')\) if \(z_{i^*}=(l_{i^*}', c_{i^*}'  \epsilon )\) for a sufficiently small \(\epsilon >0\), \(\varvec{z}_{i} = \varvec{z}_{i}'\), where \(\varvec{z}' \in \textit{PR}(e')\).

(2)
SPI. Let \(S^4\) be defined as follows. For all \(e \in \mathcal {E}\), \(\varvec{z}\in S^4(e)\) if \(\varvec{z}\) is in the Egalitarian Equivalent solution EE(e) and for all \(i, j \in N\) such that \(R_{i} = R_{j}\) and \(s_{i}=s_{j}\), \(z_{i}=z_{j}\).

(3)
IUS. Define \(S^5\) as follows. For all \(e \in \mathcal {E}\), \(\varvec{z}\in S^5(e)\) if \(\varvec{z}\in BE^{\tilde{u}}\) (other than \(\textit{EI}\)) and for all \(i, j \in N\) such that \(R_{i} = R_{j}\) and \(s_{i}=s_{j}\), \(z_{i}=z_{j}\).

(4)
NRZE. Consider the equal income Walrasian solution.

(5)
NDL. Consider \(S^6\) such that for all \(e \in \mathcal {E}\), \(S^6(e) \subseteq \textit{PR}(e)\) and \(S^6(e)=1\).
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Miyagishima, K. Implementability and equity in production economies with unequal skills. Rev Econ Design 19, 247–257 (2015). https://doi.org/10.1007/s1005801501758
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DOI: https://doi.org/10.1007/s1005801501758